Comte set out to complete the Enlightenment by extending scientific method to the last subject that had resisted it, and gave sociology both its name and its founding ambition.
His law of three stages holds that every field of knowledge passes through a theological phase, in which events are explained by will; a metaphysical phase, in which they are explained by abstractions like nature or right; and a positive phase, in which enquiry confines itself to ascertainable relations between observable phenomena and abandons the search for ultimate causes. Astronomy made the passage first, then physics, chemistry and biology; sociology is the last and the most consequential, because only it can supply the principles on which industrial society might be reordered.
The context is post-revolutionary France, and the problem as Comte saw it was not tyranny but disintegration: the old religious and monarchical order had been destroyed without anything cohering taking its place, and the critical philosophy that destroyed it was constitutionally unable to build. Positive knowledge was to do the constructive work — hence his conservative streak, and his insistence that order and progress are complements rather than opposites. The motto stands on the Brazilian flag.
He had been Saint-Simon's secretary and the break between them was bitter, but the inheritance is plain: authority passing to those who understand production and society, exercised in the name of expertise rather than birth. His late Religion of Humanity, with its calendar, sacraments and priesthood of sociologists, embarrassed his followers and was mocked by Mill, who had otherwise championed him. It was nonetheless the logical end of his position.